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Mirrors > Home > MPE Home > Th. List > opprval | Structured version Visualization version GIF version |
Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
opprval.1 | ⊢ 𝐵 = (Base‘𝑅) |
opprval.2 | ⊢ · = (.r‘𝑅) |
opprval.3 | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
opprval | ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprval.3 | . 2 ⊢ 𝑂 = (oppr‘𝑅) | |
2 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅) | |
3 | fveq2 6229 | . . . . . . . 8 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = (.r‘𝑅)) | |
4 | opprval.2 | . . . . . . . 8 ⊢ · = (.r‘𝑅) | |
5 | 3, 4 | syl6eqr 2703 | . . . . . . 7 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = · ) |
6 | 5 | tposeqd 7400 | . . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · ) |
7 | 6 | opeq2d 4440 | . . . . 5 ⊢ (𝑥 = 𝑅 → 〈(.r‘ndx), tpos (.r‘𝑥)〉 = 〈(.r‘ndx), tpos · 〉) |
8 | 2, 7 | oveq12d 6708 | . . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet 〈(.r‘ndx), tpos (.r‘𝑥)〉) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
9 | df-oppr 18669 | . . . 4 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet 〈(.r‘ndx), tpos (.r‘𝑥)〉)) | |
10 | ovex 6718 | . . . 4 ⊢ (𝑅 sSet 〈(.r‘ndx), tpos · 〉) ∈ V | |
11 | 8, 9, 10 | fvmpt 6321 | . . 3 ⊢ (𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
12 | fvprc 6223 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅) | |
13 | reldmsets 15933 | . . . . 5 ⊢ Rel dom sSet | |
14 | 13 | ovprc1 6724 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(.r‘ndx), tpos · 〉) = ∅) |
15 | 12, 14 | eqtr4d 2688 | . . 3 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
16 | 11, 15 | pm2.61i 176 | . 2 ⊢ (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
17 | 1, 16 | eqtri 2673 | 1 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∅c0 3948 〈cop 4216 ‘cfv 5926 (class class class)co 6690 tpos ctpos 7396 ndxcnx 15901 sSet csts 15902 Basecbs 15904 .rcmulr 15989 opprcoppr 18668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-res 5155 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-tpos 7397 df-sets 15911 df-oppr 18669 |
This theorem is referenced by: opprmulfval 18671 opprlem 18674 |
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